EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6415
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Geometric Properties of a General Subclass of Analytic
Functions Involving Multiplier Operator

Shasha Han1,∗, Maisarah Haji Mohd2, Mohamed Illafe1

1 Mathematics Department, School of Engineering, Mathematics and Technology, Navajo
Technical University, Crownpoint, New Mexico, United States
2 School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia

Abstract. This paper investigates a general subclass of analytic functions defined in the open
unit disk involving a multiplier transformation. Employing the linear approximation theorem, we
present sharp coefficient estimates, growth and distortion results, and radii for geometric properties
such as close-to-convexity, starlikeness, and convexity. Our approach generalizes several known
results.
2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, linear approximation, geometric properties, mul-
tiplier transformation, coefficient estimates, starlikeness

1. Introduction

Let U = {z ∈ C : |z| < 1} denote the open unit disk. Define A as the class of functions
analytic in U with the normalized form

f(z) = z +
∞∑
n=2

anz
n. (1)

We consider the subclass A∗ ⊂ A, consisting of functions with non-positive Taylor
coefficients of the form

h(z) = z −
∞∑
n=2

cnz
n, cn ≥ 0. (2)

Cho and Srivastava [1] introduced the generalized multiplier transformation given by

T r
ν f(z) = z +

∞∑
n=2

(
n+ ν

1 + ν

)r

cnz
n, r ∈ N0, ν ≥ 0. (3)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6415

Email addresses: hanshasha0532@student.usm.my (S. Han),
maisarah hjmohd@usm.my (M. Mohd), millafe@navajotech.edu (M. Illafe)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 2 of 9

This operator generalizes previous cases. For instance, when ν = 1, it corresponds to
the operator studied by Uralegaddi and Somanatha [2], and for ν = 0, it reduces to the
classical Salagean operator [3].

Yousef et al. [4] defined a general class of analytic and bi-univalent functions denoted
by Bτ

Σ(λ, µ, δ;α), characterized by the following differential inequality

ℜ

(
1− δ)

(
f(z)

z

)τ

+ δ (f(z))′
(
f(z)

z

)τ−1

+ λµz (f(z))′′
)

> α, (4)

where δ ≥ 1, τ ≥ 0, µ ≥ 0, 0 ≤ α < 1, and λ = 2δ+τ
2δ+1 .

This class has been the focus of some researchers in numerous studies addressing some
bounding problem such as Fekete-Szegö and the second Hankel determinate (see [5–11]).

In the current work, we adapt the operator T r
ν h(z) within this framework and consider

its implications in analytic functions with negative coefficients. Our goal is to define a
refined class Bτ

ν (λ, µ, δ;α) that encapsulates and extends these concepts, forming the basis
for the investigations in the following sections.

Definition 1. Let z ∈ U, and let the parameters satisfy δ ≥ 1, τ ≥ 0, µ ≥ 0, and
0 ≤ α < 1. A function f ∈ A given by (1) is said to belong to the class Bν(λ, µ, δ, τ ;α) if
the following condition holds for all z ∈ U:

ℜ

(
(1− δ)

(
T r
ν f(z)

z

)τ

+ δ (T r
ν f(z))

′
(
T r
ν f(z)

z

)τ−1

+ λµz (T r
ν f(z))

′′

)
> α, (5)

where λ = 2δ+τ
2δ+1 .

In this paper, we are generalizing the work that Illafe has done et al. in [12, 13]
by considering not to eliminate the parameter τ . Furthermore, let B∗

ν(λ, µ, δ, τ ;α) =
Bν(λ, µ, δ, τ ;α) ∩ A∗.

Lemma 1. [14] Let f(x) be a real-valued function such that |f(x)| ≪ 1, and let n ∈ R.
Then, the following linear approximation holds

(1 + f(x))n ≈ 1 + nf(x)

up to first order in f(x).

2. Coefficient Bounds and Characterization

We begin this section by establishing a characterization result that provides the nec-
essary and sufficient conditions for a function to belong to the class B∗

ν(λ, µ, δ, τ ;α). We
begin this section by establishing a characterization result that provides the necessary and
sufficient conditions for a function to belong to the class B∗

ν(λ, µ, δ, τ ;α).



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 3 of 9

Theorem 1. Let h ∈ A∗ be defined by the expansion

h(z) = z −
∞∑
n=2

cnz
n.

Then h belongs to the class B∗
ν(λ, µ, δ, τ ;α) if and only if it satisfies the condition:

∞∑
n=2

[
(τ − δ + nδ + λµn(n− 1)

](n+ ν

1 + ν

)r

cn ≤ 1− α. (6)

Proof. From the class condition (5) and applying Lemma 1, we can write

ℜ

{
(1− δ)

(
T r
ν h(z)

z

)τ

+ δ (T r
ν h(z))

′
(
T r
ν h(z)

z

)τ−1

+ λµz(T r
ν h(z))

′′

}

= ℜ

{
1 +

∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

cnz
n−1

}
> α.

Taking the limit as z → 1− along the real axis yields the validity of inequality (6). On
the other hand, assume inequality (6) holds, to prove h belongs to class B∗

ν(λ, µ, δ, τ ;α),
we need to show that for every z ∈ U that

∣∣∣∣∣(1− δ)

(
T r
ν h(z)

z

)τ

+ δ (T r
ν h(z))

′
(
T r
ν h(z)

z

)τ−1

+ λµz (T r
ν h(z))

′′ − 1

∣∣∣∣∣
=

∣∣∣∣∣
∞∑
n=2

(τ − δ + nδ + λµn(n− 1))

(
n+ ν

1 + ν

)r

cnz
n−1

∣∣∣∣∣
≤

∞∑
n=2

(τ − δ + nδ + λµn(n− 1))

(
n+ ν

1 + ν

)r

cn|z|n−1

≤
∞∑
n=2

(τ − δ + nδ + λµn(n− 1))

(
n+ ν

1 + ν

)r

cn ≤ 1− α.

This completes the proof.

Corollary 1. Suppose h(z), as defined in equation (2), belongs to the class B∗
ν(λ, µ, δ, τ ;α).

Then

cn ≤ 1− α

[τ − δ + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r , n ≥ 2. (7)

The bound (7) is sharp and attained by the function

h(z) = z − 1− α

[τ − δ + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r zn.



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 4 of 9

3. Growth and Distortion Theorems

In this section, we obtain upper and lower bounds for |h(z)| and |h′(z)| when h belongs
to the class B∗

ν(λ, µ, δ, τ ;α).

Theorem 2. Let h(z) = z −
∑∞

n=2 cnz
n be in the class B∗

ν(λ, µ, δ, τ ;α). Then, for |z| =
t < 1, the following bounds hold

|h(z)| ∈
[
t−At2, t+At2

]
, (8)

where

A =
1− α

(τ + δ + 2λµ)(2+ν
1+ν )

r
.

The bound given by inequality (8) is sharp, and equality is attained for the function

f(z) = z −Az2.

Proof. Let h(z) ∈ B∗
ν(λ, µ, δ, τ ;α) be defined by (2). Assume |z| = t < 1. Then,

|h(z)| ≤ |z|+
∞∑
n=2

cn|z|n ≤ t+ t2
∞∑
n=2

cn.

Then,
∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

cn ≤ 1− α

equivalently,

∞∑
n=2

cn ≤ 1− α

[τ − δ + nδ + λµn(n+ 1)]
(
n+ν
1+ν

)r
≤ 1− α

(τ + δ + 2λµ)
(
2+ν
1+ν

)r .
Hence,

|h(z)| ≤ t+
1− α

(τ + δ + 2λµ)
(
2+ν
1+ν

)r t2
Similarly, we can apply the same argument for the lower bound

|h(z)| ≥ |z| −
∞∑
n=2

cn|z|n ≥ t− 1− α

(τ + δ + 2λµ)
(
2+ν
1+ν

)r t2.
Therefore, the estimate in Theorem 2 follows.



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 5 of 9

Theorem 3. Let h(z) be in the class B∗
ν(λ, µ, δ, τ ;α). Then, for |z| = t < 1, the h′(z)

satisfies the inequality
1−Bt ≤ |h′(z)| ≤ 1 +Bt, (9)

where

B =
2(1− α)

(τ + δ + 2λµ)
(
2+ν
1+ν

)r .
The bounds are sharp, and equality in (9) is attained by the function

h(z) = z − 1− α

(τ + δ + 2λµ)
(
2+ν
1+ν

)r z2.
Proof. A similar argument as in the proof of Theorem 3.1 can be applied here by

considering the derivative of f .

4. Closure Properties

In this section, we establish that the class B∗
ν(λ, µ, δ, τ ;α) is closed under convex com-

binations and averaging. This follows naturally from the linearity of the operator T r
ν h(z)

and the sub-additive behavior of the defining inequality.

Theorem 4. Let hj(z) = z −
∑∞

n=2 cnjz
n ∈ B∗

ν(λ, µ, δ, τ ;α) for j = 1, 2, . . . , N . Then the
average function

H(z) :=
1

N

N∑
j=1

hj(z) = z −
∞∑
n=2

 1

N

N∑
j=1

cnj

 zn

also belongs to the class B∗
ν(λ, µ, δ, τ ;α).

Proof. Let bn = 1
N

∑N
j=1 cnj . Then, using linearity and convexity of the modulus

∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

bn

=
∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r
 1

N

N∑
j=1

cnj


=

1

N

N∑
j=1

( ∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

cnj

)

≤ 1

N

N∑
j=1

(1− α) = 1− α

Hence, H ∈ B∗
ν(λ, µ, δ, τ ;α).



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 6 of 9

Theorem 5. The class B∗
ν(λ, µ, δ, τ ;α) is convex for any h1, h2 ∈ B∗

ν(λ, µ, δ, τ ;α) and any
0 ≤ ω ≤ 1, the function

T (z) := ωh1(z) + (1− ω)h2(z)

also belongs to B∗
ν(λ, µ, δ, τ ;α).

Proof. Let

hj(z) = z −
∞∑
n=2

cnjz
n, j = 1, 2

and define

T (z) = z −
∞∑
n=2

[ωcn1 + (1− ω)cn2] z
n.

As before, the inequality becomes

∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

[wcn1 + (1− w)cn2]

= w

∞∑
n=2

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

cn1

+ (1− w)
∞∑
n=3

[τ − δ + nδ + λµn(n− 1)]

(
n+ ν

1 + ν

)r

cn2

≤ w(1− α) + (1− w)(1− α) = 1− α

Thus T ∈ B∗
ν(λ, µ, δ, τ ;α).

5. Radii of Close-to-Convexity, Starlikeness, and Convexity

Let us now derive the radii within which a function h ∈ B∗
ν(λ, µ, δ, τ ;α) exhibits the

standard geometric behaviors of close-to-convexity, starlikeness, and convexity.
Let A denote the class of normalized analytic functions in the unit disk U. For 0 ≤

β < 1, we define the following important subclasses:

• The class of close-to-convex functions of order β

C(β) =
{
h ∈ A : Re

{
h′(z)

}
> β

}
• The class of starlike functions of order β

S∗(β) =

{
h ∈ A : Re

{
zh′(z)

h(z)

}
> β

}
• The class of convex functions of order β

K(β) =

{
h ∈ A : Re

{
1 +

zh′′(z)

h′(z)

}
> β

}



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 7 of 9

In the following, we aim to determine these properties for functions belonging to the class
B∗
ν(λ, µ, δ, τ ;α).

Close-to-Convexity Radius

Theorem 6. Let h ∈ B∗
ν(λ, µ, δ, τ ;α). Then h ∈ C(β) in the disk |z| < r1, where

r1 = inf
n≥2

(1− β) [(τ − δ) + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r
n(1− α)


1/(n−1)

. (10)

Proof.
For h(z) = z −

∑∞
n=2 cnz

n, we have

h′(z) = 1−
∞∑
n=2

ncnz
n−1.

To ensure ℜ{h′(z)} > β, it is sufficient that

|h′(z)− 1| ≤ 1− β for z ∈ U.

That is,
∞∑
n=2

ncnr
n−1 ≤ 1− β.

Using the coefficient bound property represented by Theorem 1, we get

n|z|n−1 ≤
(1− β)[τ − δ + nδ + λµn(n− 1)]

(
n+ν
1+ν

)r
1− α

|z| ≤ inf
n≥2

(1− β) [(τ − δ) + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r
n(1− α)


1/(n−1)

, (11)

which yields the radius in (10).

Starlikeness Radius

Theorem 7. If h ∈ B∗
ν(λ, µ, δ, τ ;α), then h is starlike of order β in the disk |z| < r2,

where

r2 = inf
n≥2

(1− β) [(τ − δ) + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r
(n− β)(1− α)


1/(n−1)

. (12)



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 8 of 9

Proof. Using similar argument in the previous theorem, we can write∣∣∣∣zf ′(z)

f(z)
− 1

∣∣∣∣
=

∣∣∣∣∑∞
n=2(n− 1)anz

n−1

1−
∑∞

n=2 anz
n−1

∣∣∣∣ ≤ ∑∞
n=2(n− 1)an|z|n−1

1−
∑∞

n=2 an|z|n−1
≤ 1− β

(n− β)|z|n−1 ≤
(1− β) [τ − δ + nδ + λµn(n− 1)]

(
n+ν
1+ν

)r
1− α

|z| ≤ inf
n≥2

(1− β) [(τ − δ) + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r
(n− β)(1− α)


1/(n−1)

. (13)

Convexity Radius

Theorem 8. If h ∈ B∗
ν(λ, µ, δ, τ ;α), then h is convex of order β in the disk |z| < r3, where

r3 = inf
n≥2

(1− β) [(τ − δ) + nδ + λµn(n− 1)]
(
n+ν
1+ν

)r
n(n− β)(1− α)


1/(n−1)

. (14)

Proof. Following arguments analogous to those of Theorems 5.1 and 5.2, we obtain
the expression for the radius given in (14).

6. Conclusion

In this paper, we introduce a new subclass of analytic functions B∗
ν(λ, µ, δ, τ ;α), which

is defined by a generalized multiplier operator T r
ν . We derive sharp coefficient estimates,

establish growth and distortion theorems, and determine the radii of close-to-convexity,
starlikeness, and convexity. The proposed class unifies and generalizes several well-known
subclasses in geometric function theory. Potential directions for future work include ex-
ploring inclusion relationships, neighborhood properties, and the behavior of partial sums
within this class.

Conflict of Interest

The authors declare that there are no conflicts of interest.



S. Han, M. H. Mohd, M. Illafe / Eur. J. Pure Appl. Math, 18 (3) (2025), 6415 9 of 9

References

[1] N. E. Cho and H. M. Srivastava. Argument estimates of certain analytic functions de-
fined by a class of multiplier transformations. Mathematical and Computer Modelling,
37(1–2):39–49, 2003.

[2] B. A. Uralegaddi and C. Somanatha. Certain classes of univalent functions. In Current
Topics in Analytic Function Theory, pages 371–374, 1992.

[3] G. S. Sălăgean. Subclasses of univalent functions. In Complex Analysis: Fifth
Romanian-Finnish Seminar, pages 362–372. Springer, 1981.

[4] F. Yousef, S. Alroud, and M. Illafe. New subclasses of analytic and bi-univalent
functions endowed with coefficient estimate problems. Analysis and Mathematical
Physics, 11:1–12, 2021.

[5] M. Illafe, A. Amourah, and M. Haji Mohd. Coefficient estimates and fekete–szegö
inequalities for a subclass of analytic and bi-univalent functions. Axioms, 11(4):147,
2022.

[6] M. Illafe, M. H. Mohd, F. Yousef, and S. Supramaniam. A subclass of bi-univalent
functions defined by a symmetric q-derivative operator and gegenbauer polynomials.
European Journal of Pure and Applied Mathematics, 17(4):2467–2480, 2024.

[7] M. Illafe, F. Yousef, M. H. Mohd, and S. Supramaniam. Initial coefficients estimates
and fekete–szegö inequality problem for a general subclass of bi-univalent functions
defined by subordination. Axioms, 12(3):235, 2023.

[8] M. Illafe and A. A. Amourah. Coefficient estimates and fekete–szegö functional
inequalities for a certain subclass of analytic and bi-univalent functions. Axioms,
11(4):147, 2022.

[9] A. Amourah and M. Illafe. A comprehensive subclass of analytic and bi-univalent
functions associated with subordination. Palestine Journal of Mathematics, 9(1):187–
193, 2022.

[10] F. Yousef, S. Alroud, and M. Illafe. A comprehensive subclass of bi-univalent functions
associated with chebyshev polynomials of the second kind. Analysis and Mathematical
Physics, 26:329–339, 2019.

[11] M. Illafe, M. H. Mohd, and F. Yousef. Bounds for the second hankel determinant
of a general subclass of bi-univalent functions. International Journal of Mathematics
Engineering and Management Science, 9:1226–1239, 2024.

[12] M. Illafe, M. H. Mohd, F. Yousef, and S. Supramaniam. Investigating inclusion,
neighborhood, and partial sums properties for a general subclass of analytic functions.
International Journal of Neutrosophic Science, 25(3):501–510, 2024.

[13] M. Illafe, F. Yousef, M. H. Mohd, and S. Supramaniam. Fundamental properties of a
class of analytic functions defined by a generalized multiplier transformation operator.
International Journal of Mathematics and Computer Science, 19(4):1203–1211, 2024.

[14] J. Stewart. Tayler and Maclaurin Series. Cengage Learning, 8th edition, 2016.